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Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces. (English) Zbl 1123.53033

From the author’s abstract: “A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic metric with conical singularities of positive singular curvature on a compact surface of genus greater than one. The author proves that these metrics are actually realized by exactly one convex Fuchsian polyhedron (up to global isometries). This extends a famous theorem of A. D. Alexandov”.

Let ${M}_{K}^{+}$ be the simply connected (Riemannian) space of dimension 3 of constant curvature $K$, $K\in \left\{-1,0,1\right\}$. A convex polyhedron is an intersection of half-spaces of ${M}_{K}^{+}$. The number of half-spaces may be infinite, but the intersection is asked to be locally finite: each face must be a polygon with a finite number of vertices, and the number of edges at each vertex must be finite. A polyhedron is a connected union of convex polyhedra. A polyhedral surface is the boundary of a polyhedron and a convex polyhedral surface is the boundary of a convex polyhedron. A convex (polyhedral) cone in ${M}_{K}^{+}$ is a convex polyhedral surface with only one vertex. (The sum of the angles between the edges is in the interval $\left(0,2\pi \right)$.)

A metric of curvature $K$ with conical singularities with positive singular curvature on a compact surface $S$ is a (Riemannian) metric of constant curvature $K$ on $S$ minus $n$ points $\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ such that the neighborhood of each ${x}_{i}$ is isometric to the induced metric on the neighborhood of the vertex of a convex cone in ${M}_{K}^{+}$. The ${x}_{i}$ are called the singular points. By definition the set of singular points is discrete, hence finite since the surface is compact. An invariant polyhedral surface is a pair $\left(P,F\right)$, where $P$ is a polyhedral surface in ${M}_{K}^{+}$ and $F$ a discrete group of isometries of ${M}_{K}^{+}$ such that $F\left(P\right)=P$ and $F$ acts freely on $P$. The group $F$ is called the acting group. If there exists an invariant polyhedral surface $\left(P,F\right)$ in ${M}_{K}^{+}$ such that the induced metric on $P/F$ is isometric to a metric $h$ of curvature $K$ with conical singularities on a surface $S$, one says that $P$ ealizes the metric $h$ (the singular points of $h$ correspond to the vertices of $P$, and $F$ is isometric to the fundamental group of $S$). In this case one says that $h$ is realized by a unique invariant polyhedral surface $\left(P,F\right)$ if $P$ is unique up to isometries of ${M}_{K}^{+}$.

Let $P$ be the boundary of a convex compact polyhedron in ${M}_{K}^{+}$. The induced metric on $P$ is isometric to a metric of constant curvature $K$ with conical singulartities of positive curvature of the sphere. A famous theorem of A. D. Alexandrov asserts that each such metric on the sphere is realized by the boundary of a unique convex compact polyhedron of ${M}_{K}^{+}$ [see A. D. Alexandrov, Convex polyhedra, Berlin: Springer (2005; Zbl 1067.52011); H. Busemann, Convex surfaces, New York-London: Interscience Publishers (1958; Zbl 0196.55101); A. V. Pogorelov, Extrinsic geometry of convex surfaces. Providence, R.I.: American Mathematical Society (AMS) (1973; Zbl 0311.53067)] in this case the acting group is the trivial one.

In the present paper the author proves the following theorem. Theorem 1.1. A hyperbolic metric with conical singularities of positive singular curvature on a compact surface $S$ of genus $>1$ is realized by a unique convex Fuchsian polyhedron in hyperbolic space (up to global isometries). The author mentions that the idea to use the Fuchsian group of hyperbolic space comes from [M. Gromov, Partial differential relations, Berlin etc.: Springer (1986; Zbl 0651.53001)]. The general outline of the proof of Theorem 1.1 is classical and has been used in several other cases, starting from A. D. Alexandov’s works. Roughly speaking, the idea is to endow both the space of convex Fuchsian polyhedra with $n$ vertices and the space or corresponding metrics with a suitable topology, and to show that the map from one to the other given by the induced metric is a homeomorphism. The difficult step is to show the local injectivity of the map ‘nduced metric. This is equivalent to a statement on infinitesimal rigidity of convex Fuchsian polyhedra.

Reviewer: Ioan Pop (Iaşi)
##### MSC:
 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov) 52A55 Spherical and hyperbolic convexity 52B70 Polyhedral manifolds 53C24 Rigidity results (differential geometry)